Definition:Rank (Linear Algebra)
Definition
Linear Transformation
Let $\phi$ be a linear transformation from one vector space to another.
Let the image of $\phi$ be finite-dimensional.
Then its dimension is called the rank of $\phi$ and is denoted $\rho \left({\phi}\right)$.
Matrix
Let $K$ be a field.
Let $\mathbf A$ be an $m \times n$ matrix over $K$.
Then the rank of $\mathbf A$, denoted $\rho \left({\mathbf A}\right)$, is the dimension of the subspace of $K^m$ generated by the columns of $\mathbf A$.
That is, it is the dimension of the column space of $\mathbf A$.
These definitions can be proved compatible (by viewing a matrix as a lin. transform., and maybe vice versa); such is a typical PW entry. Also, cf. Definition:Finite Rank Operator
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